JournalsjemsVol. 23, No. 11pp. 3497–3520

The rôle of Coulomb branches in 2D gauge theory

  • Constantin Teleman

    UC Berkeley, USA
The rôle of Coulomb branches in 2D gauge theory cover
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I give a simple construction of the Coulomb branches C3,4(G;E){\mathscr{C}_{3,4}(G;E)} of gauge theory in three and four dimensions, defined by H. Nakajima [Adv. Theor. Math. Phys. 20 (2016)] and A. Braverman, M. Finkelberg and H. Nakajima [Adv. Theor. Math. Phys. 22 (2018)] for a compact Lie group GG and a polarizable quaternionic representation EE. The manifolds C(G;0){\mathscr{C}(G;\mathbf{0})} are abelian group schemes over the bases of regular adjoint GC{G_\mathbb{C}}-orbits, respectively conjugacy classes, and C(G;E){\mathscr{C}(G;E)} is glued together over the base from two copies of C(G;0){\mathscr{C}(G;\mathbf{0})} shifted by a rational Lagrangian section εV{\varepsilon_V}, representing the Euler class of the index bundle of a polarization V{V} of E{E}. Extending the interpretation of C3(G;0){\mathscr{C}_3(G;\mathbf{0})} as “classifying space” for topological 2D gauge theories, I characterize functions on C3(G;E){\mathscr{C}_3(G;E)} as operators on the equivariant quantum cohomologies of M×V{M\times V}, for compact symplectic G{G}-manifolds M{M}. The non-commutative version has a similar description in terms of the Γ{\Gamma}-class of V{V}.

Cite this article

Constantin Teleman, The rôle of Coulomb branches in 2D gauge theory. J. Eur. Math. Soc. 23 (2021), no. 11, pp. 3497–3520

DOI 10.4171/JEMS/1071